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pseudosphere wikipedia 119 captures 30 may 2004 23 sep 2026 oct nov dec 14 2021 2022 2023 success fail about this capture timestamps the wayback machine http web archive org web 20221114053354 https en wikipedia org wiki pseudosphere pseudosphere from wikipedia the free encyclopedia jump to navigation jump to search geometric surface in geometry a pseudosphere is a surface with constant negative gaussian curvature a pseudosphere of radius r is a surface in r 3 displaystyle mathbb r 3 having curvature 1 r 2 in each point its name comes from the analogy with the sphere of radius r which is a surface of curvature 1 r 2 the term was introduced by eugenio beltrami in his 1868 paper on models of hyperbolic geometry 1 contents 1 tractroid 2 universal covering space 3 hyperboloid 4 see also 5 references 6 external links tractroid edit tractricoid the same surface can be also described as the result of revolving a tractrix about its asymptote for this reason the pseudosphere is also called tractroid as an example the half pseudosphere with radius 1 is the surface of revolution of the tractrix parametrized by 2 t t tanh t sech t 0 t displaystyle t mapsto left t tanh t operatorname sech t right quad quad 0 leq t infty it is a singular space the equator is a singularity but away from the singularities it has constant negative gaussian curvature and therefore is locally isometric to a hyperbolic plane the name pseudosphere comes about because it has a two dimensional surface of constant negative gaussian curvature just as a sphere has a surface with constant positive gaussian curvature just as the sphere has at every point a positively curved geometry of a dome the whole pseudosphere has at every point the negatively curved geometry of a saddle as early as 1693 christiaan huygens found that the volume and the surface area of the pseudosphere are finite 3 despite the infinite extent of the shape along the axis of rotation for a given edge radius r the area is 4π r 2 just as it is for the sphere while the volume is 2 3 π r 3 and therefore half that of a sphere of that radius 4 5 universal covering space edit the pseudosphere and its relation to three other models of hyperbolic geometry the half pseudosphere of curvature 1 is covered by the interior of a horocycle in the poincaré half plane model one convenient choice is the portion of the half plane with y 1 6 then the covering map is periodic in the x direction of period 2 π and takes the horocycles y c to the meridians of the pseudosphere and the vertical geodesics x c to the tractrices that generate the pseudosphere this mapping is a local isometry and thus exhibits the portion y 1 of the upper half plane as the universal covering space of the pseudosphere the precise mapping is x y v arcosh y cos x v arcosh y sin x u arcosh y displaystyle x y mapsto big v operatorname arcosh y cos x v operatorname arcosh y sin x u operatorname arcosh y big where t u t t tanh t v t sech t displaystyle t mapsto big u t t operatorname tanh t v t operatorname sech t big is the parametrization of the tractrix above hyperboloid edit in some sources that use the hyperboloid model of the hyperbolic plane the hyperboloid is referred to as a pseudosphere 7 this usage of the word is because the hyperboloid can be thought of as a sphere of imaginary radius embedded in a minkowski space see also edit hilbert s theorem differential geometry dini s surface gabriel s horn hyperboloid hyperboloid structure quasi sphere sine gordon equation sphere surface of revolution references edit beltrami eugenio 1868 saggio sulla interpretazione della geometria non euclidea treatise on the interpretation of non euclidean geometry gior mat in italian 6 248 312 also beltrami eugenio opere matematiche mathematical works in italian vol 1 pp 374 405 isbn 1 4181 8434 9 beltrami eugenio 1869 essai d interprétation de la géométrie noneuclidéenne treatise on the interpretation of non euclidean geometry annales de l école normale supérieure in french 6 251 288 archived from the original on 2016 02 02 retrieved 2010 07 24 bonahon francis 2009 low dimensional geometry from euclidean surfaces to hyperbolic knots ams bookstore p 108 isbn 0 8218 4816 x chapter 5 page 108 stillwell john 2010 mathematics and its history revised 3rd ed springer science business media p 345 isbn 978 1 4419 6052 8 extract of page 345 le lionnais f 2004 great currents of mathematical thought vol ii mathematics in the arts and sciences 2 ed courier dover publications p 154 isbn 0 486 49579 5 chapter 40 page 154 weisstein eric w pseudosphere mathworld thurston william three dimensional geometry and topology vol 1 princeton university press p 62 hasanov elman 2004 a new theory of complex rays ima j appl math 69 521 537 doi 10 1093 imamat 69 6 521 issn 1464 3634 archived from the original on 2013 04 15 stillwell j 1996 sources of hyperbolic geometry amer math soc london math soc henderson d w taimina d 2006 experiencing geometry euclidean and non euclidean with history aesthetics and mathematics pdf springer verlag kasner edward newman james 1940 mathematics and the imagination simon schuster pp 140 145 155 external links edit non euclid crocheting the hyperbolic plane an interview with david henderson and daina taimina norman wildberger lecture 16 history of mathematics university of new south wales youtube 2012 may pseudospherical surfaces at the virtual math museum retrieved from https en wikipedia org w index php title pseudosphere oldid 1114532969 categories differential geometry hyperbolic geometry surfaces spheres hidden categories cs1 italian language sources it cs1 french language sources fr articles with short description short description is different from wikidata navigation menu personal tools not logged in talk contributions create account log in namespaces article talk english views read edit view history more search navigation main page contents current events random article about wikipedia contact us donate contribute help learn to edit community portal recent changes upload file tools what links here related changes upload file special pages permanent link page information cite this page wikidata item print export download as pdf printable version in other projects wikimedia commons languages български català deutsch español français 한국어 ido italiano עברית nederlands polski română русский slovenščina svenska türkçe українська 中文 edit links this page was last edited on 7 october 2022 at 00 20 utc text is available under the creative commons attribution sharealike license 3 0 additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia mobile view developers statistics cookie statement